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I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not sure how to derive the explicit representation from the weight system deformed by extended Dynkin diagram?

To decompose a irreducible representation under a maximal regular subalgebra, we look at the extended Dynkin diagram, which has one more node from the minus highest root added on the original Dynkin diagram, and we then eliminate another node to deform the extended Dynkin diagram into the Dynkin diagram for the maximal regular subalgebra. The key point is to replace the Dynkin coefficient corresponding to the eliminated node by the one corresponding to the extra node of the minus highest root. Thus, deforming the weight system of the given irrep with above replacement gives us the branching rules and the Cartan matrices for the subalgebra in the given irrep. For example, under the subgroup $SU(2)\times SU(2)\times SU(2) \Subset SO(7)$, we have the 8 dimensional irrep of SO(7) deformed as $8 \rightarrow (1,2,2) + (2,1,2)$, see P33-34 LieART.

To construct the explicit matrix representation, we define the coordinates for each simple roots in an orthonormal basis (Cartan-Weyl basis) and therefore the coordinates for the fundamental weights as well. Thus, the generators corresponding to Cartan matrices have its matrix entries as the coordinate of every weights. Now, my question is what are the fundamental weights for the deformed Dynkin diagram? Should I keep the old fundamental weights except the replaced one or the new fundamental weights are resolved from the deformed Dynkin diagram? Most importantly, what's the new fundamental weight corresponding to the extended root?

I would also need help to reveal how generators of the subalgebra are in the subset of generators of the mother algebra. When I replace the Dynkin coefficient with the extended root, how could the corresponding Cartan matrix belong to linear combination of Cartan matrices of the mother algebra?

I'm a Physics student with poor knowledge in Lie algebra. Please reinterpret my description with proper language whenever you needed to.

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I don't quite understand what you mean by "explicit matrix elements". At first sight it suggests that you want to describe representations with explicit matrices, which is hard because the matrices may be huge (the size of the dimension of the representation), and require choosing an explicit basis in the representation. However from the rest of the question I gather that you are really just interested in computing weights of representations, which is much easier (and less dependent on basis choices).

For branching in general (you describe the special case of branching to equal rank maximal semisimple subalgebras) you need as basic ingredient to know how the fundamental weights of the full group are expressed in terms of fundamental weight of the subalgebra. The fundamental weights are the dual basis to the simple coroots, where the latter are the functions $\lambda\mapsto\frac{\langle\lambda,\alpha\rangle}{\langle\alpha,\alpha\rangle}$ for $\alpha$ a simple root. If you exchange one root for another one so as to still form a simple system of roots (in the current case that root is minus the highest root) then only the corresponding coroot changes, but through the relation between dual bases, this changes all fundamental weights. However, you have the fortunate circumstance that you do not need to know the fundamental weights for the subalgebra, bit only to express the fundamental weights for the full algebra in them. But that means just apply the simple coroots for the subalgebra to the fundamental weights of the full algebra. Since most coroots have not changed, you matrix of evaluations will in large part be just the identity matrix, only the one row coming from the new coroot needs to be different: it is the expression of the coroot for minus the highest root as linear combination of the original simple coroots.

The above assumes the new simple root just takes the place of the removed simple root. In practice you will probably want to number the root system for the subalgebra in a way that is usual for systems of that type, and this may involve some permutation of the coordinates.

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